[r-t] Proportion of Surprise Methods

[Richard Smith]

Richard Smith wrote:

> with 1456 now allowed at the first division end, and 1236 at the second, 
> these change to about 1/9 to be TB, 4/9 to be S, and 5/9 to be D.

Spot the typo.  Obviously the last fraction should have been 
4/9 too.

> If we go further and allow up to four consecutive blows in one place, we 
> would expect these ratios to remain approximately the same.  And this is 
> indeed what happens. There are 378,916 grids in total, of which 47% (179,941) 
> are S, 10% (36,721) are TB and 43% (162,254) are D.

And if anyone's interested what happens when we allow an 
arbitrary number of consecutive blows in one place, there 
are 1,166,400 methods of which 518,400 (that is, precisely 
4/9 of the total) are Delight, another 518,400 are Surprise, 
and the remaining 129,600 are Treble Bob.  This works out 
exactly because there are no longer any restrictions that 
prevent the division-end changes from being selected 
independently.  All that remains are the following 
restrictions:

  - standard treble dodging treble path;
  - normal parity structure (sufficient, though not
    necessary, for producing an extent);
  - palindromic

(Requiring a normal parity structure together with a 
palindromic lead implicitly requires a double change at LE 
and HL.)

1,166,400 is 1080 squared, though the methods are not simply 
every overwork coupled with every underwork.  This is 
because of the requirement to keep a normal parity structure 
in the middle division (i.e. when the treble is in 3-4). 
Specifically, if we have (say) x5x over the treble, we could 
not have x2x (the same thing upside-down) under the treble: 
we would need either 222 or xxx under.

RAS